Download Natural Human Mobility Patterns and Spatial Spread of Infectious

PHYSICAL REVIEW X 1, 011001 (2011)
Natural Human Mobility Patterns and Spatial Spread of Infectious Diseases
Vitaly Belik,1,* Theo Geisel,1,2 and Dirk Brockmann3,4
1
Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
2
Faculty of Physics, University of Göttingen, Göttingen, Germany
3
Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois, USA
4
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois, USA
(Received 13 October 2010; revised manuscript received 23 May 2011; published 8 August 2011)
We investigate a model for spatial epidemics explicitly taking into account bidirectional movements
between base and destination locations on individual mobility networks. We provide a systematic analysis
of generic dynamical features of the model on regular and complex metapopulation network topologies
and show that significant dynamical differences exist to ordinary reaction-diffusion and effective force of
infection models. On a lattice we calculate an expression for the velocity of the propagating epidemic
front and find that, in contrast to the diffusive systems, our model predicts a saturation of the velocity with
an increasing traveling rate. Furthermore, we show that a fully stochastic system exhibits a novel threshold
for the attack ratio of an outbreak that is absent in diffusion and force of infection models. These insights
not only capture natural features of human mobility relevant for the geographical epidemic spread, they
may serve as a starting point for modeling important dynamical processes in human and animal
epidemiology, population ecology, biology, and evolution.
DOI: 10.1103/PhysRevX.1.011001 Subject Areas: Complex Systems, Interdisciplinary Physics, Statistical Physics
The geographic spread of emergent infectious diseases,
epitomized by the 2009 H1N1 outbreak and subsequent
pandemic [1], the worldwide spread of SARS in 2003
[2,3], and recurrent outbreaks of influenza epidemics
[4–6], is determined by a combination of disease relevant
human interactions and mobility across multiple spatial
scales [7]. While infectious contacts yield local outbreaks
and proliferation of a disease in single populations, multiscale human mobility is responsible for spatial propagation
[8]. Therefore, a prominent lineage of mathematical models has evolved that is based on reaction-diffusion dynamics [9,10] in which the combination of local exponential
growth and diffusive dispersal captures qualitative aspects
of observed dynamics. Typically these systems exhibit
constant velocity epidemic wave fronts. A related class
of phenomenological models is based on the concept of
an effective force of infection across distance and thus does
not require explicit modeling of dispersal [11,12].
Although more sophisticated models [2,4,13,14] have
been developed to describe the dynamics of recent emergent epidemics such as the H1N1 pandemic or SARS,
taking into account long distance travel and multiscale
mobility networks [15], the majority of these models are
still based on the interplay of local reaction kinetics and
*[email protected]
Present address: Massachusetts Institute of Technology,
Department of Civil and Environmental Engineering,
Cambridge, MA, USA.
Published by the American Physical Society under the terms of
the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and
the published article’s title, journal citation, and DOI.
2160-3308=11=1(1)=011001(5)
diffusion processes on networks of metapopulations. The
key assumptions of diffusive transport [Fig. 1(a)] are that
(a) individuals behave identically, (b) movements are stochastic, (c) spatial increments are local and as consequence
individuals eventually visit every location in the system.
Although it is intuitively clear that these assumptions are
idealizations and in conflict with everyday experience, the
difficulty is how to refine them when data on mobility is
lacking, insufficient, or incomplete. Fortunately, a series of
recent studies [16–19] substantially advanced our knowledge on multiscale human mobility. An important discovery that emerged from these studies are individual mobility
networks, i.e., individuals typically only visit a limited
number of places frequently, predominantly performing
commutes between home and work locations and possibly
a few other locations. Consequently, individuals or groups
of individuals exhibit spatially constrained movement patterns despite their potentially high mobility rate. It has
remained elusive how this novel and important empirical
insight on individual mobility networks can be reconciled
with epidemiological models, to what extent it may impact
spatial disease dynamics, and how it may alter spreading
scenarios and predictions promoted by ordinary reactiondiffusion models, in which mobile hosts can reach every
location in the system.
In this paper we demonstrate how individual mobility
networks can be incorporated into a class of models for
spatial disease dynamics, and address the questions of
whether and how significantly the existence of spatially
constrained individual mobility networks impacts on key
features of disease dynamics. To this end we investigate a
model that explicitly accounts for the bidirectional mobility of individuals between their unique base location
011001-1
Published by the American Physical Society
PHYS. REV. X 1, 011001 (2011)
VITALY BELIK, THEO GEISEL, AND DIRK BROCKMANN
populations is diffusive dispersal among those populations
defined by hopping rates wnm > 0 from population m to n,
which yields a metapopulation reaction-diffusion system,
i.e., for infected and susceptible individuals,
X
@t In ¼ !Sn In =Nns # "In þ ðwnm Im # wmn In Þ;
@t Sn ¼
FIG. 1. Metapopulational mobility models. Patches and arrows
represent individual populations and travel. (a) Diffusive dispersal: indistinguishable individuals travel randomly between
different locations governed by the set of transition rates !nm .
(b) Dispersal capturing individual mobility patterns: individuals
with label k possess travel rates !kmk and !kkm at which they
travel from their base location k to connected locations m and
back.
(e.g. their home) and a small set of other locations
[Fig. 1(b)]. The entire population is therefore represented
by a set of overlapping individual mobility networks associated with each base location, see, e.g., [20–22]. We focus
on the analysis of epidemics on regular lattices and complex metapopulation networks and systematically compare
the dynamics to reaction-diffusion systems as well as the
force of infection models. In conflict with reactiondiffusion models, in which front velocities increase with
travel rates unboundedly, we show that our model predicts
a saturation of wave front velocities, a direct consequence
of the rank of locations in individual mobility patterns.
This suggests that estimates for propagation speeds may
have been considerably overestimated in the past.
Furthermore, we analyze a fully stochastic model to
show that both, in regular lattices as well as complex
metapopulation networks, the global outbreak of a disease
is determined by a novel type of threshold for the attack
ratio of the disease which depends on the characteristic
time spent at distant locations. Finally, we show that in the
limit of low and high travel rates our model agrees with
reaction-diffusion models and the direct force of the infection class of models, respectively.
Consider a system of M populations labeled m and
assume that in each an epidemic outbreak can be described
by a compartmental SIR model, i.e.,
!
Im þ Sm !2Im ;
"
Im !Rm ;
(1)
where Sm , Im , and Rm label and quantify susceptible,
infected, and recovered individuals of population m.
Infections and recovery events occur at rates ! and ",
respectively, with ! > ". The number of individuals in
population m is given by Nm ¼ Sm þ Im þ Rm , which,
however, is conserved only in the statistical sense at equilibrium. This system of reactions yields the mean-field
descriptions @t I ¼ !IS=N # "I and @t S ¼ #!IS=N. A
natural and plausible extension to a system of M coupled
#!Sn In =Nns
Nns
þ
X
m
m
ðwnm Sm # wmn Sn Þ;
(2)
is the number of individuals in population n
where
in diffusive equilibrium. Travel rates !mn are usually
estimated by gravitylike laws [23]. Nns is determined by
detailed balance, i.e., Nns =Nms ¼ wnm =wmn and the conservation P
of the number of individuals in the metapopulation
N ¼ m Nms . These types of models have been employed
in numerous recent studies [2,14,24,25]. The tight connection to spatially continuous reaction-diffusion systems is
revealed for the special case of a linear grid of populations
placed at regular intervals l at positions xm ¼ ml and
mobility between neighboring populations, i.e., wnm ¼
!#n#1;m þ !#nþ1;m which yields
@t j ¼ !js # "j þ D@2x j;
@t s ¼ #!js þ D@2x s; (3)
in which jðx; tÞ ¼ In =Nns , sðx; tÞ ¼ Sn =Nns , and D ¼ l2 !.
Equation (3) is related to a classic form of a Fisher equation
which has been deployed in mathematical epidemiology
[9,10]. For sufficiently localized initial conditions this
systems exhibits traveling waves with speed
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi
c ¼ 2 !Dð1 # "=!Þ & !:
(4)
Note that this velocity increases monotonically with the
mobility rate !.
In order to account for individual mobility patterns we
propose the following generalization: individuals possess
two indices n and k, characterizing their current and their
base location, respectively. The dispersal dynamics is governed by a Markov process
!kmn
k;
Xnk Ð Xm
!knm
(5)
where X stands for each infectious state S, I, and R.
Equation (5) implies that individuals with base location k
possess a unique mobility rate wknm that determines how
they travel between locations n and m. This yields a generalization of (2)
X
! kX m
k # !k I k Þ;
@t Ink ¼
Sn In # "Ink þ ð!knm Im
mn n
Nn m
m
(6)
X
! X
@t Skn ¼ # Skn Inm þ ð!knm Skm # !kmn Skn Þ;
Nn m
m
where Ink and Skn are the number of infected and susceptible
individuals of type k that are currently located in population
Nn is the total number of individuals at n, i.e., Nn ¼
P n.
k
k
k
k
k ðIn þ Sn þ Rn Þ. If !nm are k independent, we recover
the reaction-diffusion case described above. In the
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NATURAL HUMAN MOBILITY PATTERNS AND SPATIAL . . .
following we consider the case of overlapping star-shaped
networks corresponding to commuting between base and
destination locations only. This implies that !knm ¼ 0 if
either k ! n or k ! m. We further assume the constant
return rate wkmk ¼ w# for all k and m. Realistically we
have !nkn =!# ( 1, implying individuals belonging to n
remain at their base most of the time.
If mobility rates are large compared to the rates associated with the infection and recovery, i.e., !kmk , !# ) !,
", the detailed balance condition is fulfilled for infected
and susceptible individuals separately and the last term in
Eq. (6) vanishes in which case the above model is equivalent to the remote-force-of-infection model—see also
[11,21].
In general, it is difficult, if not impossible, to extract the
dynamic differences between the systems defined by
Eqs. (2) and (6) for an arbitrary metapopulation. Yet, it is
crucial to understand the key dynamic differences that
emerge when individual mobility patterns replace ordinary
diffusion. We therefore investigate the impact of individual
mobility patterns in the instructive one-dimensional lattice
case. We assume that only infected individuals can travel
and that recovery is absent (SI model), i.e., " ¼ 0 (relaxing
these two restrictions does not change the main results but
clarifies the analysis). We denote the number of infected
individuals with base location n that are located on site n
(i.e. their base) and sites n * 1 by Inn and In* , respectively.
At rates !þ and !# individuals leave and return to their
base. In order to obtain a spatially continuous description
* ! I * ðx * lÞ + I * * lrI * þ l2 !I *
we approximate In*1
2
and introduce relative concentrations un ¼ In =N, vn ¼
ðInþ þ In# Þ=N leading to
@t v ¼ 2!þ u # !# v;
the forward rate !þ . For a systematic comparison to the
ordinary reaction-diffusion system, we have to establish a
relation between !* and ! in Eq. (4). For the sake of
simplicity we consider the symmetric case, i.e., !þ ¼
!# ¼ !, yielding
pffiffiffiffiffiffiffiffi
2 6D!!
:
(9)
c¼
ð! þ 3!Þ
Numerical simulations of a fully stochastic system nicely
agree with the analytical predictions (Fig. 2). The essential
pffiffiffiffi
feature of cð!Þ is its saturation as ! ! 1 whereas c & !
for reaction-diffusion systems. This effect is a direct consequence of the spatial constraints imposed by individual
mobility patterns, i.e., increasing the mobility rate ! yields
a higher frequency of travel events but is restricted to the set
of locations connected to the base location and thus is
universal and also holds for more complex metapopulation
topologies [26]. For high rates, the deviation between ordinary reaction-diffusion and bidirectional mobility increases
without bounds. This is an important insight as the expression for wave front speeds for reaction-diffusion have been
used in the past to estimate wave front speeds from mobility
rates and vice versa. Our results indicate that if constrained
mobility patterns and bidirectional movement patterns are
at work these estimates must be treated with particular care.
Note that in the limit ! ! 1 Eq. (7) reduces to the heuristic
equation in Ref. [27].
A key question in epidemiological contexts concerns
conditions under which an epidemic outbreak propagates
or wanes. Outbreak dynamics are usually triggered by
crossing thresholds inherent in the system’s dynamics
[28]. For example the basic reproduction number R0 of
a disease, i.e., the expected number of secondary cases
(7)
where D ¼ l2 =2. The function uðx; tÞ is the fraction of
individuals based on and located at x whereas vðx; tÞ is the
fraction of individuals based on but not located at x.
Note that we discarded here a third equation for w ¼
ðI þ # I# Þ=N independent from (7) with a solution
vanishing at long times. This does not change the main
result (8). The nontrivial steady state is given by
u? ¼ !# =ð2!þ þ !# Þ, v? ¼ 2!þ =ð2!þ þ !# Þ with
u? þ v? ¼ 1 as expected.
One of the key questions is if the system (7) exhibits
stable propagating waves as solutions and how their velocity depends on system parameters. The ansatz uðx; tÞ ¼
u~ðx # ctÞ, and likewise for v, leads to a stable solution with
a velocity given by
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2!!þ Dð2 þ !# =!þ Þ
c¼
:
(8)
ð! þ !# þ 2!þ Þ
Fixing !# , this implies that c ! 0 as !þ ! 0. On the other
hand, if !# is small, the system is entirely determined by
5
4
Front velocity, c
@t u ¼ !ð1 # u # vÞðu þ v þ D!vÞ þ !# v # 2!þ u;
PHYS. REV. X 1, 011001 (2011)
SI
SI
3
SIR
SI (immobile S)
2
1
0
0
2
4
6
8
10
Travel rate, ω
FIG. 2. Front velocity cð!Þ of the model defined by Eq. (6) as
a function of mobility rate ! in comparison to ordinary reactiondiffusion dynamics [Eq. (2)]. For stochastic simulations we used
a lattice with 102 sites and N ¼ 104 individuals/site. Mean
velocities are indicated by red (bidirectional mobility) and
blue (reaction-diffusion) symbols. Analytical results, i.e.,
Eqs. (9) and (4) are shown by dashed and dash-dotted lines,
respectively.
011001-3
VITALY BELIK, THEO GEISEL, AND DIRK BROCKMANN
caused by a single infected individual in a susceptible
population represents such a threshold [8]. In the single
population SIR dynamics introduced above R0 ¼ !=". If
R0 > 1, an outbreak occurs, otherwise the epidemic dies
out. In the extended metapopulation system with diffusive
host movements a second threshold, known as the global
invasion threshold [24], is induced by the flux of individuals between populations, which must be sufficiently high
for an epidemic to spread throughout the metapopulation.
An interesting and important feature of the model we
propose is the existence of another invasion threshold
which is determined by the return rate !# or equivalently
by the time an individual spends outside the base location.
This finding is illustrated in Fig. 3, which depicts the attack
ratio $ as a function of !# for a bidirectional SIR epidemic
on a lattice. The attack ratio $ is defined as the fraction of
the overall population affected during an epidemic. We
fixed the entire interlocation flux at a value above the
global invasion threshold ensuring a global outbreak in
the reaction-diffusion system.
This result is universal for the bidirectional system and
is independent of the particular topology. Insets (a) and
(b) of Fig. 3 display simulation results for uncorrelated
scale-free and Erdős-Rényi networks. For low return rates
the attack ratio is close to unity. However, with growing
values of the return rate, the attack ratio decreases steeply
and vanishes, reflecting the absence of the global outbreak.
The regime of high return rates corresponds to small
dwelling times on distant locations. Infected individuals
do not have enough time to transfer the disease to susceptibles in unaffected locations before they return. Note that
1
1
0.8
Attack ratio, ρ
(b)
N=2000
0.5
N=1000
N=500
0.6
1
0
−1
10
2
10
N=200
0.5
0
−1
10
1
10
(a)
0.4
0.2
0
10
N=100
0
−1
10
0
10
1
10
0
10
N=50
2
10
1
10
2
10
Travel rate, ω−/N
FIG. 3. Results of stochastic simulations. Attack ratio
$ð!# =NÞ as a function of the return travel rate !# for SIR
epidemics on a lattice with 102 sites with N agents/site.
Epidemic parameters: ! ¼ 1, " ¼ 0:1. Insets: (a) $ð!# =NÞ
for a scale-free network (& ¼ 1:5) of 103 nodes populated
uniformly with hNi ¼ 250=site, with kmin ¼ 5 and kmax ¼ 50;
(b) $ð!# =NÞ for an Erdős-Rényi network with 500 nodes and
k" ¼ 10. Results were averaged over 50 realizations.
Interlocation flux rate was h!i ¼ 1.
PHYS. REV. X 1, 011001 (2011)
for a wide range of local population sizes (50–2000) the
attack ratio $ collapses onto a universal curve $ &
$ð!# =NÞ which suggests that the basic mechanism of
the invasion threshold and its functional dependence on
return rate !# can be understood theoretically.
To estimate the observed invasion threshold on a onedimensional lattice for R0 * 1 we calculated the number
of infected individuals % that originate from an affected
location and can seed an outbreak in an unaffected one
[28]. These seeders are approximately given by the total
number of individuals entering the new location per unit
time, i.e., N! multiplied by the typical time they remain
active in the destination location. This time is given by the
inverse rate r of exiting the seeders class. Since two
processes (returning to the base location and recovery)
can trigger this exit, r ¼ " þ !# . Therefore, % +
N!=ð" þ !# Þ. In the tree approximation the threshold
condition for an epidemic on a network with an average
degree k" is given by %ðk" # 1ÞðR0 # 1Þ > 1 [24]. Thus, for
a one-dimensional lattice the threshold condition reads
#
"
!#
!þ
#1 ;
(10)
& 2NðR0 # 1Þ
"
"
where we kept the total interlocation flux ! ¼ 2!þ !# =
ð2!þ þ !# Þ constant (this relation and positivity of travel
rates impose the restrictions: !# > ! and !þ < !=2) and
used !þ ( !# . Moreover, as extensive simulations show,
the global invasion threshold in terms of the flux rate !
exists in bidirectional systems only for low return rates !# .
With increasing return rate the disease fails to propagate
globally, even for constant interlocation flux !. This effect,
observed in lattice topologies and paradigmatic more realistic network topologies, is a fundamental consequence of
birectional mobility patterns. This effect is absent in
reaction-diffusion systems fully determined by the total
interlocation flux. This property implies that the invasion
of a front is limited only by return rates and not as is
typically assumed by the overall particle flux !. In the
context of disease dynamics this implies that more efficient
containment strategies could be potentially devised that do
not target the overall mobility but rather, modify mobility
patterns in an asymmetric way.
Note that the transition takes place only in a system with
a finite number of agents per site; in the system with an
infinite number of individuals this effect would disappear,
as also confirmed in Fig. 3. Indeed from Eq. (10) for
" ( !# the scaling $ & $ðN!
!# Þ follows—with an increasing number of individuals per site N, the threshold value of
the return rate !# increases.
Recently, an unprecedented amount of detailed information on human activity became available, requiring the
revision of established models and the development of
new more sophisticated ones. We investigate an epidemiological model explicitly incorporating such important properties of human mobility as individual mobility networks
and frequent bidirectional movements between home and
011001-4
NATURAL HUMAN MOBILITY PATTERNS AND SPATIAL . . .
destination locations. It manifests surprising dynamical
features such as the existence of bounded front velocity
and a novel propagation threshold, which are universal for
any metapopulation topology. As more detailed data continue to become available our approach is a promising
foundation for further research.
D. B. and V. B. acknowledge support from the
Volkswagen Foundation, D. B. acknowledges EU-FP7
grant Epiwork. V. B. and D. B. would like to thank W.
Noyes and H. Schlämmer for invaluable comments.
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